More on Bounded Independence Plus Noise: Pseudorandom Generators for Read-Once Polynomials

We construct pseudorandom generators with improved seed length for several classes of tests. First, we consider the class of read-once polynomials over GF(2) in m variables. For errors we obtain seed length (O) over tilde (log(m/epsilon) log (1/epsilon)). This is optimal up to a factor of log(1/epsilon) . poly log log (m/epsilon). The previous best seed length was polylogarithmic in m and 1/epsilon. Second, we consider product tests f : {0,1}(m) -> C-<= 1. These tests are the product of k functions f(i ): {0,1}(l) C-<= 1, where the inputs of the f, are disjoint subsets of the m variables and C-<= 1 is the complex unit disk. Here we obtain seed length l. polylog(m/epsilon). This implies better generators for other classes of tests. If moreover the f(i) have output range {-1, 0, 1} then we obtain seed length (O) over tilde((log(k/epsilon) + l) (log (1/epsilon) + log log m)). This is again optimal up to a factor of log(1/epsilon) . polylog(l, log k, logm, log(1/epsilon)), while the previous best seed length was >= root k. A main component of our proofs is showing that these classes of tests are fooled by almost d-wise independent distributions perturbed with noise.